Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(1,2)(3,6)(4,8)(5,7)(12,21)(15,20)(17,24)(19,22);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,22)(13,18)(15,24)(16,23)(17,20)(19,21)Polycyclic presentation:
2^7|a^2=g,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,d^b=df,e^b=eg,d^c=dg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1640, #18485, #18520, … |
| maximal normal subgroup of | C2×#1640, #18485, #18520, … |
| maximal quotient of | #7770, #7773, #7782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 372);